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Systems of equations: elimination

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Solve the system: x + 2y = 7 and 3x + 2y = 13.

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Subtract the first equation from the second to eliminate y. What is the result?

Common mix-up

In this example, a student wrote 2x = 13, so x = 13/2.

The student subtracted the left sides but not the right sides, writing 2x = 13 instead of 2x = 6.

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See why this works

How does subtracting the equations remove a variable?

When both equations share a term like 2y, that term becomes your exit: subtract the equations and it disappears.

Matching coefficients let you add or subtract the equations so that one variable cancels.

Look for matching terms when you face a system like x + 2y = 7 and 3x + 2y = 13. Both equations contain 2y. If the same amount appears in each one, subtracting one whole equation from the other removes it entirely, because the same quantity minus itself is zero.

Dry-erase diagram: Systems of equations: elimination — Matching coefficients let you add or subtract the equations so that one variable cancels.
See the ideaSystems of equations: elimination: Matching coefficients let you add or subtract the equations so that one variable cancels.

Subtract carefully, left side minus left side, right side minus right side. The left side gives 3x + 2y - (x + 2y) = 2x, with the y terms vanishing. The right side gives 13 - 7 = 6. So the new equation is 2x = 6. Forgetting the right side and writing 2x = 13 is the most common slip here.

Solve 2x = 6 by dividing by 2 to get x = 3. One variable is now known, so the original system can be finished by back-substitution. Replace x in x + 2y = 7, giving 3 + 2y = 7. Subtract 3 and divide by 2, and y = 2.

Both equations must hold for the final pair. With x = 3 and y = 2, the first equation reads 3 + 4 = 7 and the second reads 9 + 4 = 13. Elimination looks like a shortcut, but it is really just adding the same thing to equal things, so balance is preserved at every line.

Keep this idea

Line up the equations, add or subtract so one variable cancels, solve, then back-substitute into either original equation.

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